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Four-wave mixing processes are ubiquitous in ultrafast optics and thedetermination of the coefficients of the $\chi^{(3)}$ tensor is thus essential.We introduce a novel time-resolved ultrafast spectroscopic method tocharacterize the third-order nonlinearity on the femtosecond time-scale. Thisapproach, coined as "nonlinear chirped Doppler interferometry", makes use ofthe variation of the optical group delay of a transmitted probe under theeffect of an intense pump pulse in the nonlinear medium of interest. Theobservable is the spectral interference between the probe and a reference pulsesampled upstream. We show that the detected signal is enhanced when the pulsesare chirped, and that, although interferometric, the method is immune toenvironmental phase fluctuations and drifts. By chirping adequately thereference pulse, the transient frequency shift of the probe pulses is alsodetected in the time domain and the detected nonlinear signal is enhanced.Nonlinear phase shifts as low as 10\,mrad, corresponding to a frequency shiftof 30\,GHz, i.e. 0.01\% of the carrier frequency, are detected withoutheterodyne detection or active phase-stabilization. The diagonal and/ornon-diagonal terms of reference glasses (SiO$_2$) and crystals (Al$_2$O$_3$,BaF$_2$, CaF$_2$) are characterized. The method is finally applied to measurethe soft vibration mode of KTiOAsO$_4$ (KTA).

Nonlinear chirped Doppler interferometry for \c{hi}(3) spectroscopy